Making a Mathematical Statistics Course More Modern

Who Are We

slide about Jessica

Who Are We

slide about Matt

Your Turn

Motivation

What Do We Mean by “Math-Stat”?

Motivation: Student Feedback

Activity

What Our Students Said (Top 3 Response Types)

• Applicable to Future: “I defined useful as something I think I will continue to use or see again in the future—skills or ideas that expand into future learning goals.”
• Applicable to the “Real World”: “I based this on how applicable these things would be to my professional life or general knowledge after college….”
• Deepened Understanding: “I (also) considered how much something deepened my conceptual understanding.”

Is Math Stat “Useful?”

• Could math stat be more “useful?”

• JSM 2003 Session “Is the Math Stat Course obsolete?”

quotes I like: The math stat course has not changed in 40 years, whereas statistics has changed enormously, so how could the course not be obsolete? - Moore

“Leave out point estimation (or allow 5 min tops), hypothesis testing, power, type I error, type II error, and sample size calculation. Put these topics into the”learn to be a consultant” course. Leave out one sample z, two sample t, one sample , chi-square. Leave it out! There is a lot in the traditional course that is worth while but the psyche of the traditional course is not working anymore. The ways to turn it around are not that different than the statistical literacy course….. leave out UMVUE and unbiasedness.” - Reid

“The worse thing is that it tends to bore the teachers, and if teachers are bored, students don’t get the liveliness of the subject.” - Efron

idea that comes up a bit: modeling.

“We should de-emphasize t-test, jackknife, named nonparametric methods, asymptotics (I’ve never seen an infinite sample), Cramer-Rao lower bound. It should not be too much like a catalog. I would add modeling, computing (likelihood graphics), problem solving, decision calculations (how to think about the best thing to do), risk, odds, expectation, sample and experimental design (at least touch on that), foundational issues such as understanding p-values (direction of conditioning), worrying about whether they are any good.” - Morris

DM: I heard no support for continuing to teach optimal testing (Neyman-Pearson) in first course? All: right

question about proofs: “I don’t know. My first reaction is to say do not teach formal derivation. Var( x ) is reasonably accessible, but if BE did not understand the t distribution when he first saw it, then your students won’t. I can’t see emphasizing so much the derivations of these distributions, especially those using multivariate calculus. Reality has surpassed us – you will not be able to squeeze all of this into this course.”

Activity: In or Out?

Activity: Short Story

• Fostering Conceptual Understanding in Mathematical Statistics by Green and Blankenship (2015).
  • published in The American Statistician.

Activity: Short Story

A “meaningful story” is one continuous piece of writing / creative work that uses key words from a list and in which the sentences “make sense and hang together.” That is, the ideas in the story must illustrate that you understand key concepts from math-stat in a way that allows you to write “meaningfully” about them. It is your job to use the terms in a way that demonstrates that you understand the statistical concepts involved and why we care about these terms in the big picture of statistical theory.

Activity: Short Story

What Our Students Wrote About

• 80’s music.
• applying to jobs (this is a spring semester course of mostly seniors!).
• gender inequality in women’s and men’s hockey.
• living on a farm.
• all kinds of sports!

What a Student Said

“The second mini-project was the most impactful assessment for me….This project was very different from much of the Math/Stat curriculum and a unique opportunity to show understanding of different terms. Being forced to use these terms in the context of a sentence helps to develop a better understanding of what they mean. It is difficult to explain terms like estimator, variance, or binomial distribution in a statistical context of a story without a good understanding of their meanings….”

Motivation: Student Feedback

“I really hope that I remember Bayes. I feel like this would be helpful for me to remember because I want to study epidemiology and public health. As we learn more about disease and track new diseases it will probably be useful to update our ideas with what we have learned.”

“In my opinion, the process of coming up with a prior is a creative one. There is no single correct answer for a prior distribution, it is only based on previous knowledge around a subject. Different people may come up with different priors, and in turn create a posterior distribution based on data examined after the prior is created. It takes creativity to be a Bayesian”

“I would like to know more about how experts in a field develop informative priors for Bayesian Modeling.”

“Bayesian statistics is a whole different world where we think about the parameters as random quantities and not having true values. I really liked the process of starting with a prior distribution and then using the data to update our beliefs with a posterior distribution.”

“I initially struggled understanding Bayesian statistics. I could not wrap my head around that concept because I like to find an answer from a given formula. I do not like when answers can be subjective and with Bayesian statistics, the choice of the prior distribution is highly subjective. I was getting frustrated with myself because I was struggling to understand the material and how to do the homework problems. However, I had this realization one afternoon that in order to understand this material, I had to let go of the idea that there was only one correct way to set up a Bayesian problem.”

“I struggled to understand the Bayesian unit at first. I didn’t really understand why we were doing this odd method where we made up stuff to start. I finally understood how the prior distribution affected the posterior, with the parameters acting like observations. We can think that our prior beliefs are worth some actual data, which will contribute to the data we have to create a potentially more accurate model.”

Activity: Bayes

## Activity: Bayes

“One topic I found interesting was Bayesian Statistics. This topic incorporates things I have enjoyed, like kernel matching and finding likelihood functions. It was interesting going over different examples in class. I particularly enjoyed the cat-lover example. We started with a prior belief about the proportion of people who prefer cats to dogs. Then, through a show-of-hands, we collected data. This example took the words off the slides for me, demonstrating a practical application of the method. My prior belief was that the majority of people prefer dogs. However, if I remember correctly, none of us raised our hands in support of cats. Despite my prior belief, this was still super surprising.”

Activity: p-value Paper

Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “p< 0.05”. The American Statistician, 73(sup1), 1-19.

“Moving beyond ‘statistical significance’ opens researchers to the real significance of statistics, which is ‘the science of learning from data, and of measuring, controlling, and communicating uncertainty’ (Davidian and Louis 2012).”

“These p-values We’ve Been Using for 4 Years Are Flawed??”

Activity: p-value Paper

On Handout.

What Our Students Said

“I always considered p-values to be the best way to measure significance and did not question what a p-value really meant. The project taught me what statistical thinking is and to take into account more than just whether a p-value passes some threshold. In intro statistics, we are taught to follow specific steps and formulas without much background knowledge or reasoning. I am someone who does not often question information, especially in classes…..”

What Our Students Said

“(The p-value paper) was probably the most thought-provoking assignment for me because it forced me to think critically about how we conduct research. Coming from an Economics background, there is a lot of conversation around the reliability or quality of research. I felt that this assessment was the most impactful for me long-term because it forced me to think critically about something that I had previously just taken for granted….”

What Our Students Said

“I hope to remember that hypothesis tests are fragile instruments. I hope to remember that they do not prove anything and provide insight into how much evidence there is for or against the null hypothesis in a probabilistic sense.”

Activity: Assessment